Symmetry properties of sign-changing solutions to nonlinear parabolic equations in unbounded domains

TL;DR

This paper investigates the long-term symmetry properties of solutions to nonlinear parabolic equations in unbounded domains, showing that solutions tend to become axially symmetric and nonincreasing in angle over time under certain conditions.

Contribution

It establishes the asymptotic foliated Schwarz symmetry of solutions in unbounded domains, extending symmetry results to sign-changing solutions, Hénon-type problems, equilibria, and periodic solutions.

Findings

Solutions become axially symmetric over time

Sign-changing solutions exhibit foliated Schwarz symmetry

Symmetry results apply to equilibria and periodic solutions

Abstract

We study the asymptotic (in time) behavior of positive and sign-changing solutions to nonlinear parabolic problems in the whole space or in the exterior of a ball with Dirichlet boundary conditions. We show that, under suitable regularity and stability assumptions, solutions are asymptotically (in time) foliated Schwarz symmetric, i.e., all elements in the associated omega-limit set are axially symmetric with respect to a common axis passing through the origin and are nonincreasing in the polar angle. We also obtain symmetry results for solutions of H\'enon-type problems, for equilibria (i.e. for solutions of the corresponding elliptic problem), and for time periodic solutions.